Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The potential ($V$) at any axial point, at $2$ m distance ($r$) from the centre of the dipole of dipole moment vector $\vec{P}$ of magnitude, $4 \times 10^{-6}$ C m, is $\pm 9 \times 10^3$ V.
(Take $\dfrac{1}{4\pi\epsilon_0} = 9 \times 10^9$ SI units)
Reason R : $V = \pm\dfrac{2P}{4\pi\epsilon_0 r^2}$, where $r$ is the distance of any axial point, situated at $2$ m from the centre of the dipole.
In the light of the above statements, choose the correct answer from the options given below:
Answer: (A) A is true but R is false.
On the axis of a short dipole:
$$V = \pm\frac{1}{4\pi\epsilon_0}\frac{P}{r^2} = \pm\frac{9 \times 10^9 \times 4 \times 10^{-6}}{2^2} = \pm 9 \times 10^3\ \text{V}$$
So A is true (the sign depends on which side of the dipole the point is).
R has an extra factor of 2. $\dfrac{2P}{4\pi\epsilon_0 r^3}$ is the axial electric field, and the axial potential is $\dfrac{P}{4\pi\epsilon_0 r^2}$. So R is false.
Solution by Sreeraj P, M.Sc Physics